For any two statement p and q, the negative of the expression p ∨...

$~ \left(p ∨ \left(~ p ∧ q\right)\right)$

Using $~ \left(a ∨ b\right) = ~ a ∧ ~ b, ~ \left(a ∧ b\right) = ~ a ∨ ~ b \text{ and } ~ \left(~ a\right) = a,$ we get

$= ~ p ∧ ~ \left(~ p ∧ q\right)$

$= ~ p ∧ \left(~ \left(~ p\right) ∨ q\right)$

$= ∼ p ∧ \left(p ∨ ∼ q\right)$

Using distributive law, i.e. $a ∧ \left(b ∨ c\right) = \left(a ∧ b\right) ∨ \left(a ∧ c\right),$ we get

$= \left(~ p ∧ p\right) ∨ \left(~ p ∧ ~ q\right)$

Using $~ a ∧ a = C,$ where $C$ represents contradiction, we get

$= C ∨ \left(~ p ∧ ~ q\right)$

Using $C ∨ a = a,$ we get

$= ~ p ∧ ~ q .$